Example application of the IAU 2000 resolutions

concerning Earth orientation and rotation


Patrick Wallace


HMNAO/CCLRC/RAL, p.t.wallace@rl.ac.uk



(Original version 20 July 2004, revised 29 July 2004;

this reformatted and corrected version 20 January 2006)




Here is an example application of the IAU 2000 resolutions concerning Earth orientation and rotation.  The objective is to predict the topocentric apparent direction of a star.


The nomenclature scheme I am using is as follows.  Coordinates are in upper case;  transformations and corrections are in mixed case.



                 CATALOG RA,DEC


      proper motion, catalog epoch to J2000


            ICRS RA,DEC, EPOCH J2000            <- example starts here


         proper motion, J2000 to date


                  BCRS RA,DEC




               ASTROMETRIC RA,DEC


               light deflection

               annual aberration


                  GCRS RA,DEC


                   frame bias




                  CIRS RA,DEC


                 Earth rotation




                  polar motion






        diurnal aberration and parallax


               TOPOCENTRIC HA,DEC




               TOPOCENTRIC AZ,ALT               <- example ends here




                 OBSERVED AZ,ALT



The circumstances are as follows:


  Site 9.712156 deg E, 52.385639 deg N, 200 m ASL (ITRS)


  Fictitious "Tycho 2" star, epoch 2000

  RA,Dec = 353.22987757, +52.27730247 deg

  Proper motions +22.9, -2.1 mas/year

  Parallax 23 mas

  Radial velocity +25 km/s


  UTC  2003/08/26  00:37:38.973810


  IERS X,Y,DUT1 corrections: +0.038,-0.118 mas, -0.349535 s


  IERS polar motion x_p = +0.259371, y_p = +0.415573 arcsec


The results in summary form are as follows (in degrees):


  RA,Dec, ICRS epoch 2000      353.22987757000, +52.27730247000

  RA,Dec, BCRS                 353.22991549972, +52.27730034185

  RA,Dec, astrometric          353.22991889091, +52.27730584235

  RA,Dec, GCRS                 353.23789320667, +52.27695262534

  RA,Dec, CIRS                 353.23300208264, +52.29554173960

  HA,Dec, topocentric           -0.29507962185, +52.29549062657

  Az,Alt, topocentric          116.44983979538, +89.79843387822

  Az,Alt, observed             116.44983979538, +89.79848801461


BCRS stands for baryentric celestial reference system.  GCRS stands for geocentric celestial reference system.  CIRS stands for celestial intermediate reference system; this RA,Dec is the IAU 2000 counterpart to geocentric apparent place, and is referred to the celestial intermediate origin (CIO) instead of the equinox.  The coordinates HA,Dec and Az,Alt are not affected by the introduction of the IAU 2000 methods.


The key variables and some of the workings are presented below.


Before we apply the IAU 2000 celestial-to-terrestrial transformation, we must first estimate the geocentric direction of the star by applying the usual corrections for proper motion, light deflection and annual aberration.


For the example date:


  TT   52877.0268884005800000 (MJD)


we will use the following (approximate) Earth ephemeris data:


  heliocentric posn  +0.895306712607  -0.430362177777  -0.186583142292 AU

  barycentric posn   +0.898130398596  -0.433663195906  -0.188058184682 AU

  barycentric vel    +0.007714484109  +0.013933051305  +0.006040258850 AU/day


The ICRS RA,Dec at the catalog epoch (2000.0) is:


  RA,Dec, ICRS epoch 2000    353.22987757000, +52.27730247000


Applying space motion and parallax, we obtain the astrometric place - the ICRS coordinates of an infinitely distant star coincident with the

one under study:


  astrometric place          353.22991889091, +52.27730584235


The light deflection from the Sun (we will neglect the other solar-system bodies) takes us to:


  with light deflection      353.22991848163, +52.27730517509


Annual aberration produces the proper direction, which is the GCRS RA,Dec:


  with aberration            353.23789320667, +52.27695262534


This completes the preliminaries;  we are now ready to apply the IAU 2000 celestial-to-terrestrial transformations that are the purpose of this demonstration.


To predict the orientation of the Earth we can either implement the IAU resolutions literally, working via classical methods, or we can use the direct series supplied by IERS.  The former is somewhat more computationally efficient (and more instructive), the latter easier to get right.


To demonstrate the classical methods, we start with the frame bias:


  dpsi       -41.7750 mas

  depsilon    -6.8192 mas

  dRA        -14.6000 mas


giving an ICRS to mean J2000 rotation matrix of:


  +0.99999999999999420000  -0.00000007078279744199  +0.00000008056217146976

  +0.00000007078279477857  +0.99999999999999690000  +0.00000003306041454222

  -0.00000008056217380987  -0.00000003306040883981  +0.99999999999999620000


The IAU 2000 corrections to the IAU 1976 precession, for the current date, are:


  dpsi       -10.932010 mas

  depsilon    -0.920821 mas


giving an obliquity of date of:


  epsilon_A    +84379.739145661 arcsec


and ultimately a J2000-to-date classical precession matrix:


  +0.99999960442692650000  -0.00081577397935781730  -0.00035448385722876160

  +0.00081577398094000060  +0.99999966725634080000  -0.00000014012603875794

  +0.00035448385358768210  -0.00000014905272408423  +0.99999993717058570000


Using the IAU 2000A model, we obtain the classical nutation components

dpsi and depsilon:


  luni-solar            -12.687774156, +5.669802082 arcsec

  planetary              +0.000048676, +0.000119415 arcsec


which combine to give:


  nutation, IAU 2000A   -12.687725480, +5.669921497 arcsec


and the following classical mean-to-true nutation matrix:


  +0.99999999810814740000  +0.00005643620233914664  +0.00002446753280028101

  -0.00005643552974070033  +0.99999999802968630000  -0.00002748924554327810

  -0.00002446908414069593  +0.00002748786465307690  +0.99999999932284070000


Combining the rotation matrices, we obtain for the classical NxPxB matrix:


  +0.99999965722043850000  -0.00075940856976379120  -0.00032993579616347590

  +0.00075939951242126470  +0.99999971127592400000  -0.00002757624279218965

  +0.00032995664253816620  +0.00002732568025683818  +0.99999994519095910000


This matrix could be used to transform the geocentric place into the true place of date, with the CIP as the pole and the true equinox of date as the RA origin.  The transformation to terrestrial coordinates would then require the mean sidereal time and the equation of the equinoxes, complicated functions involving both UT and TT.  As we are following IAU 2000 methods, we are instead going to work via the celestial intermediate system, so that the transformation to terrestrial coordinates requires only the Earth rotation angle, a simple linear function of UT.


We extract the CIP X,Y coordinates from the matrix (elements (3,1) and



  CIP X,Y         +0.000329956642538, +0.000027325680257


(As mentioned earlier we can, if we wish, generate X,Y directly from series.  For the date in question we obtain:


  from series     +0.000329956644592, +0.000027325684592


Note the good agreement.)


If the utmost accuracy is needed, the next step is to add the small IERS corrections (which come from VLBI observations):


  IAU 2000A       +0.000329956642538, +0.000027325680257

  corrected       +0.000329956826767  +0.000027325108177


Having obtained X,Y, we compute the small quantity s using the series for s+XY/2:


  s        -2.900355 mas


From X, Y and s we can obtain the rotation matrix that transforms directions from the celestial reference system into the celestial intermediate system of date:


  +0.99999994556424450000  +0.00000000955326444293  -0.00032995682715159210

  -0.00000001856936992539  +0.99999999962666920000  -0.00002732510353706674

  +0.00032995682676736510  +0.00002732510817669448  +0.99999994519091400000


Earlier, we computed the RA,Dec in the celestial reference system at which the star appears:


  GCRS         353.23789320667, +52.27695262534


The rotation matrix transforms this into the celestial intermediate system:


  CIRS         353.23300208264, +52.29554173960


We are now ready to move from coordinates on the celestial sphere into coordinates on the Earth.  This involves Earth rotation together with three small effects:


  diurnal aberration     << 1 arcsec

  s'                     << 0.1 mas

  polar motion           << 1 arcsec


Because these effects are small, the precise way they are applied is of little consequence;  individual applications thus have some leeway. For example, in some applications it may be more convenient to deal with the diurnal aberration at the same time as annual aberration and to eliminate the geocentric stage completely.  In a similar way, some applications will need the star direction in terms of the geographical coordinates of the sub-star point whereas others (and this example) will express the same information in the form of a local hour angle and declination.


In all cases, the Earth rotation angle is the main element in the transformation.  The UT1 is:


  UT1  52877.02614148466 (MJD)


and the corresponding ERA is:


  ERA  343.2256920994647 degrees


The IERS Conventions set out the CIRS-to-ITRS transformation as a large rotation about the z-axis corresponding to the ERA, followed by small z, y and x rotations that take into account s' and polar motion.


The tiny quantity s' for the given date is:


  s'     -0.001714681 mas


The rotation matrix for the s' and polar motion portion is:


  +0.99999999999920940000  -0.00000000000831300878  +0.00000125746609283028

  +0.00000000001084649458  +0.99999999999797040000  -0.00000201475475899438

  -0.00000125746609281098  +0.00000201475475900642  +0.99999999999717980000


Allowing for diurnal aberration, Earth rotation, site longitude and polar motion produces the following local HA,Dec:


  HA,Dec, topocentric     -0.29507962185, +52.29549062657


or, rotated into local horizon coordinates:


  Az,Alt, topocentric    116.44983979538, +89.79843387822


For this site and time, the star has just passed almost overhead.  Note that this is the "topocentric" rather than "observed" position;  if pointing a real telescope or antenna, the next correction would be for atmospheric refraction.


Note that some numbers in this report are quoted to a number of decimal places beyond that corresponding to the floating-point precision of the computer used to generate them.  Repeating the calculations on a different platform may produce slightly different results.